Convergence complexity analysis of Albert and Chib's algorithm for Bayesian probit regression

TL;DR

This paper thoroughly analyzes the convergence complexity of Albert and Chib's Bayesian probit regression algorithm, showing it converges geometrically with bounds that hold as sample size and covariates grow, and provides computable bounds on convergence.

Contribution

It introduces a novel convergence analysis using drift and minorization conditions, avoiding common pitfalls, and establishes bounds that hold in high-dimensional settings.

Findings

Convergence rate bounded below 1 as sample size increases with fixed p.

Convergence rate bounded below 1 as p increases with fixed sample size.

Provides the first computable bounds on total variation distance to stationarity.

Abstract

The use of MCMC algorithms in high dimensional Bayesian problems has become routine. This has spurred so-called convergence complexity analysis, the goal of which is to ascertain how the convergence rate of a Monte Carlo Markov chain scales with sample size, $n$, and/or number of covariates, $p$. This article provides a thorough convergence complexity analysis of Albert and Chib's (1993) data augmentation algorithm for the Bayesian probit regression model. The main tools used in this analysis are drift and minorization conditions. The usual pitfalls associated with this type of analysis are avoided by utilizing centered drift functions, which are minimized in high posterior probability regions, and by using a new technique to suppress high-dimensionality in the construction of minorization conditions. The main result is that the geometric convergence rate of the underlying Markov chain is bounded below 1 both as $n \rightarrow \infty$ (with $p$ fixed), and as $p \rightarrow \infty$ (with $n$ fixed). Furthermore, the first computable bounds on the total variation distance to stationarity are byproducts of the asymptotic analysis.